Class 12 Physics Top 10 Must-Know 5-Mark Derivations with Proofs (CBSE & BSEB 2026/2027)

Consider an infinitely long thin wire having uniform linear charge density $\lambda$. By cylindrical symmetry, choose a coaxial Gaussian cylinder of radius $r$ and length $l$.

Examiner Tip: State explicitly that the flux through the circular flat caps is zero because $\vec{E} \perp d\vec{A}$ ($\cos 90^\circ = 0$).

Let a thin spherical shell of radius $R$ carry total charge $Q = 4\pi R^2 \sigma$. Construct a concentric spherical Gaussian surface of radius $r$.

Plates of area $A$ separated by distance $d$. In vacuum thickness $(d-t)$, electric field is $E_0 = \frac{\sigma}{\varepsilon_0}$. Inside dielectric of thickness $t$, field is $E = \frac{E_0}{K}$.

Free electrons under electric field $E$ experience acceleration $a = -eE/m$. With mean relaxation time $\tau$, drift velocity is $v_d = \frac{eE\tau}{m}$.

Radius $R$, current $I$, point $P$ at axial distance $x$. Resolving $d\vec{B}$ into components; perpendicular components cancel by symmetry:

Consider a rectangular Ampèrian loop $abcd$ of length $L$. The magnetic field outside is negligibly weak ($B \approx 0$), and along perpendicular segments $bc$ and $da$, $\vec{B} \perp d\vec{l}$.

Two coaxial solenoids $S_1$ (radius $r_1$, turns $n_1$) and $S_2$ (radius $r_2 > r_1$, turns $n_2$). When current $I_2$ flows in $S_2$, field inside is $B_2 = \mu_0 n_2 I_2$.

For single convex spherical refracting surface: $\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}$. Applying this across two surfaces of a thin lens in air ($\mu_1 = 1, \mu_2 = \mu$):

A plane wavefront $AB$ is incident at angle $i$ on interface separating medium 1 (speed $v_1$) from medium 2 (speed $v_2$). Time taken for secondary wavelet to travel from $B$ to $C$ is $\tau = BC / v_1$.

An astronomical telescope consists of an objective ($f_o$, large aperture) and an eyepiece ($f_e$, small aperture).